\magnification=\magstep1

$$
\eqalign{
{\rm KE}_i \ + \ {\rm GPE}_i \quad &= \quad {\rm KE}_f \ + \ {\rm GPE}_f \cr
  {{1}\over{2}} m v_i^2 \ + \ 0 \quad &= \quad 0 \ + \ mgH \cr
}
$$

$$
\quad \Rightarrow \qquad H \quad = \quad {{v_i^2}\over{2g}} 
$$


$$
\eqalign{
{\rm KE}_i \ + \ {\rm GPE}_i \quad &= \quad {\rm KE}_f \ + \ {\rm GPE}_f \cr
  {{1}\over{2}} m v_i^2 \ - \ {{G M_E m}\over{R_i}} \quad &= 
                  \quad 0 \ - \ {{G M_E m}\over{R_f}} \cr
}
$$

$$
{{1}\over{2}} m v_i^2 \ - \ {{G M_E m}\over{R_E}} \quad = 
                  \quad 0 \ - \ {{G M_E m}\over{\infty}} 
$$

$$
v_i \quad = \quad \sqrt{ {{2 G M_E}\over{R_E}} }
$$

$$
v_{\rm esc} \quad = \quad \sqrt{ {{2 G M}\over{R}} }
$$

$$
R_{BH} \quad = \quad { {{2 G M}\over{c^2}} }
$$


$$
{\rm GPE} \quad = \quad - \ {{G M m}\over{R}}
$$




\bye
